498 ON THE ESTIMATION OF 



Denoting by a the radius of the circular area with- 

 in which his skill would, on the average of an im- 

 mense number of shots, enable him to plant half the 

 total number discharged; and by M the fraction ex- 

 pressing the probabiHty in question, certainty being ex- 

 pressed by I, we shall have 



while for H the prohaUlHy of hitting the same area 



we have 



H= \M 



(3.) From these expressions, knowing the value of ^, 

 which is the inverse measure of the skill of the shooter 

 (being less the greater that skill), it is easy to calculate 

 his chance of hitting a circle of any given radius in a 

 single shot. And, reversing the question, his skill 

 (measured by the fraction j) may be ascertained, by 

 observing what percentage of shots he can plant, on a 

 large average, from a given distance, within a circle of 

 any given radius (r). For that percentage being the 

 numerical expression of his probability of hitting the 

 circle, or the value of ZT, or i M, M is known, and a 

 will be given by the formula. 



I Z(7^. 2 I Lo^ s;. 2 



^= ^- V~ Log. M. ^' '^'' I^g.{iH) 



Thus, if a marksman be observed to plant 9 per 

 cent, of his arrows within a circle of one foot in 

 diameter at the distance of one hundred yards, we have 



